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# Higher Derivative Corrections to the Abelian Born-Infeld Action using Superspace Methods

Faculteit: Science and Bio-engineering Sciences

D

2.20

All the efforts done in theoretical physics today can grosso modo be split into

two categories. The first is finding the fundamental underlying principles that

govern physical phenomena and the second is trying to explain what we observe

in nature starting from these principles. The quest for a theory of everything is

primordially an effort find a single underlying principle. At the moment there

are two complementary theories which combined give an excellent description of

nature: the Standard Model and General Relativity. From this it is apparent

that more work is needed towards a single underlying theory. In the ideal case

we need a theory that unifies gravity with the other forces and correctly predicts

the twenty-six free parameters in the Standard Model. A promising candidate is

String Theory in which one assumes that elementary particles are small vibra-

ting strings rather than point-particles. In the low energy regime, the resulting

model is essentially a supersymmetric version of general relativity coupled to a

supersymmetric gauge theory.

Although string theory has many pleasing features it also is spawned with

various connected problems. Efforts to solve these naturally opened up a the door

to the discovery of D-branes which revolutionized string theory. A Dp-brane is a

p-dimensional dynamical object defined by the fact that open strings can end on

it. A tantalizing aspect of D-branes is their intimate relation with gauge theories.

Indeed, the effective action of a single brane in the slowly varying field limit is

the Born-Infeld action which in leading order is nothing more than the action

for ordinary Maxwell theory. This is the result not including derivatives on the

fieldstrength of the U(1) gauge field. It was shown that the term containing

two derivatives vanishes and it was not until recently that the four derivative

corrections where calculated.

In this thesis we study two-dimensional supersymmetric non-linear sigma-

models with boundaries. We derive the most general family of boundary condi-

tions in the non-supersymmetric case. Next we show that no further conditions

arise when passing to the N = 1 model and we present a manifest N = 1 off-shell

formulation. The analysis is greatly simplified compared to previous studies and

there is no need to introduce non-local superspaces nor to go (partially) on-shell.

Whether or not torsion is present does not modify the discussion. Subsequently,

we determine under which conditions a second supersymmetry exists. As for the

case without boundaries, two covariantly constant complex structures are needed.

However, because of the presence of the boundary, one gets expressed in terms of the other one and the remainder of the geometric data. Finally we recast some

of our results in N = 2 superspace.

Leaning on these results we then calculate the beta-functions through three

loops for an open string sigma-model in the presence of U(1) background. Re-

quiring them to vanish is then reinterpreted as the string equations of motion

for the background. Upon integration this yields the low energy effective action.

Doing the calculation in N = 2 boundary superspace significantly simplifies the

calculation. The one loop contribution gives the effective action to all orders

in alpha’ in the limit of a constant fieldstrength. The result is the well known

Born-Infeld action. The absence of a two loop contribution to the beta-function

shows the absence of two derivative terms in the action. Finally the three loop

contribution gives the four derivative terms in the effective action to all orders in

alpha’. Modulo a field redefinition we find complete agreement with the proposal

made in the literature. By doing the calculation in N = 2 superspace, we get a

nice geometric characterization of UV finiteness of the non-linear sigma-model:

UV finiteness is guaranteed provided that the background is a deformed stable

holomorphic bundle.